What is the significance of Murphy's law?

I have to contradict Ankit's otherwise theoretically sound answer, be it just because it is a bit too negative for such an optimistic movie. In the first minutes of the movie Cooper at least gives us his and his late wife's own interpretation of Murphy's Law (and the reason why he named his daughter after it), which I would extrapolate to also be the movie's preferred interpretation:

Murphy's Law doesn't mean that something bad will happen. It means that whatever can happen will happen.

And in this way he pretty much takes this on the surface rather pessimistic epigram and turns it into an optimistic anything-is-possible attitude which fits to his pioneering personality of believing in the future and trying to make things possible, which I think goes congruent with the movie's themes. Or to quote him once again

We will find a way, we always have.

And we did afterall!

And if we take a look into the interview with the Nolan brothers conducted by Jordan Goldberg (as printed in the book Interstellar: The Complete Screenplay with Selected Storyboards), we can get some more insights about the significance of Murphy's Law for the movie as viewed by screenwriter Jonathan Nolan, who seems to largely agree with Cooper and me:

JN Yeah, it's kind of a lot of my speculative bullshit. What was interesting about Murphy's Law was that I thought it was a fascinating jumping-off point for a theme --

CN Well, it's not speculative bullshit! You mean in terms of it meaning whatever can happen, will happen?

JN No, that's not speculative bullshit.

CN The other end of it is complete bullshit, and Kip has explained that to me. I left it in the film because I thought it was really good. [...] The scientific idea that Brand expounds that because Miller's and Mann' planets orbit a black hole that not enough elements are able to reach them. Kip went with it at first, but then he talked to some of his colleagues and he's fairly confident that, in fact, the opposite would be true...

JN ...The idea that I was really trying to get to with Murphy's Law was just that we live in a space in which things are permitted to happen, right? And those are good things or bad things. Death, destruction, but also life. The idea that we're here on this ball of mud hurtling through space seems to have something to do with our interactions with extra-planetary bodies and cosmic rings --

And they arrive back at Murphy's Law when discussing the importance of space exploration for the progress of humanity, its significance in our own history and our future footprints in the universe at large:

JN ...That drive to get out, to explore the universe, will be the residue that's left behind. Armstrong will be the person that people talk about. This is what connects it back, for me, to Murphy's Law. Murphy's Law popularly is: if something could go wrong, it will go wrong. The eponymous Murphy was one of a group of rocket engineers working for the Air Force. [...] And one day he came back and said, 'You know, if there was a way for these guys to mess this up, they would.' And that has been how we remember Murphy's Law, right? If there's a way for human beings to mess things up, they will. But another guy there -- an engineer by the name of Nichols -- had a different take on it. And by his accounting, the sentiment [...] was that 'If something can happen, it will happen.' It doesn't have to be a bad thing; it doesn't have to be a good thing. Humans are capable of both. We're lucky enough to live in a universe in which both things can happen. And it's interesting to see how many films and books tend to concentrate on all the terrible things that we can do. The parasisitc relationship between innovation and violence and all those sorts of things. And there's something under it, certainly. But they do tend to ignore the self-evident part too. [...] We exist. We survived. Hopefully, we will continue to survive, and that is probably a good thing.

Which also suggests a rather open-minded and neutral if not positive attitude towards Murphy's Law, concentrating on the fact that not only the bad things are determined to happen but also the good things and our existence is proof that we have overcome all odds until now and will likely overcome future odds, too. Our demise is not more likely than our persistence against it.

Murphy's law is an adage or epigram that is typically stated as: Anything that can go wrong will go wrong. The way is is said in the movie is a quote by mathematician Augustus De Morgan which he gave on June 23, 1866, analyzing Murphy's law

"The first experiment already illustrates a truth of the theory, well confirmed by practice, whatever can happen will happen, if we make enough trials."

Now how do we perceive this? Let's start by a few examples Although Murphy's law is applicable in a number of fields, I can only cite examples from my field i.e. Software engineering. While the examples might not make much sense to folks outside this field of work, I will try to keep the language simple:

1. Software Programming : computer programmers are first compiled and then run. The programmer has control of the program while it is compiled (any number of times). The programmer can make any number changes in the code to make sure that the program will run correctly. However, once the program is running, the programmer has no control whatsoever. So if programmer has not analyzed a part deeply, in other words has not taken all sorts of precautions, a bug might appear at runtime, given enough number of trials which is testing in Software industry. A very good example of this is the introduction of Generics in Java 5.0, whose sole purpose is to reduce the risk at runtime and do all that is possible at compile time, because anything that can go wrong will go wrong, given enough number of trials.

Second example:

2. How Google makes the huge amount of data available to the world

To manage the enormous amount of content on the Internet, Google uses GFS i.e. Google File System. Not going into deep details, let me just tell you the core principle behind this. Any data/content on Internet is stored on a computer which is also called server. Now what will happen to the data if the computer goes down. The manufacturer may say "I have made it so well, there are rare chances that it will fail." So either you can rely on this or rely on Murphy's law. Google engineers follow Murphy's law and lay the principle that at the end of the day, it's just a machine so however good it might be, there are chances even 0.1% that it can fail. So they think that if it can fail, it will fail and hence you can never rely on any data being kept on a single machine. So they make multiple replicas of same data and that's how they make sure everything will always work.

This is my interpretation of Murphy's law. Now in relation with movie's plot, my inference is this :

The human race on earth, (even today) knows that resources are being depleted at fast rate, and population is rising at faster rate. We also know that species before us, were wiped out from earth in a million years time, may be.

So since earth, may one day become inhabitable for humans, it definitely will become inhabitable at one time. In the initial few minutes of the movie, we see people living on earth which is already in a pretty bad state. Still they are hoping that somehow earth will be saved and humans will be able to sustain on earth. No one even acknowledges the need to abandon earth. Common perception is "how to save earth". What they are not acknowledging is that earth can not be saved. It will end, one day. The dialogue from older Dr. Brand sums it up when he says "We are not supposed to save earth. We are supposed to leave it." or as the tagline for the movie goes "Human race was born on earth. It was never meant to die on it."

Earth can end one day and hence it will end one day.

This is Murphy's law.